Towards Identifiable Representations under Misspecified Structure

Yuke Li; Yujia Zheng; Ziyi Chen; Kun Zhang; Heng Huang

cs.LGSep 27, 2026

Towards Identifiable Representations under Misspecified Structure

Authors: Yuke Li, Yujia Zheng, Ziyi Chen, Kun Zhang, Heng Huang

Organizations: University of Maryland College Park, College Park, MD, USA · Carnegie Mellon University, Pittsburgh PA, USA

Abstract

The presence of noise that depends on the latent variables poses a fundamental challenge to identifiability. Existing results rely on conditional independence among the observations given the latent variables. We study a more general \emph{misspecified structure}, where this conditional factorization does not hold, and establish both precise and approximate identifiability guarantees. We characterize structural misspecification as a perturbed factor analysis problem. For precise identifiability, we establish subspace identifiability under spectral separation and controlled perturbation, followed by component-wise identifiability under structural sparsity. When the precise condition is not guaranteed, we derive an approximate subspace-identifiability theorem. Based on these results, we develop an unsupervised variational estimator for recovering latent variables. Experiments demonstrate the effectiveness of the proposed framework.

Figures & tables

Figure 1: Visualization of the data generations of Eq. 1 under misspecified structure.

Figure 2: The overall framework of our proposed approach consists of: (1) two encoders $q_{\psi}$ and $q_{\phi}$ that map observations ${\mathbf{x}}_{t}$ to $\hat{{\mathbf{z}}}$ and $\hat{\epsilon}$ , respectively; (2) a decoder that reconstructs observations $\hat{{\mathbf{x}}}$ from $\hat{{\mathbf{z}}}$ and $\hat{\epsilon}$ ; and (3) two prior estimation modules $p_{\delta}$ and $p_{\gamma}$ that models the prior of ${\mathbf{z}}$ and $\epsilon$ , respectively. We train the framework by $L_{\text{Recon}}$ along with $L_{\text{KLD}}$ .

Figure 3: (a) Visualization of the correlations between each component of true latent variables ( ${\mathbf{z}}^{i}$ ) and their corresponding component of estimated latent variables ( $\hat{{\mathbf{z}}}^{i}$ ) using our approach. The green bounding boxes highlight the components that are identified. (b) Mean Correlation Coefficient (MCC) scores comparing our framework with state-of-the-art approaches, including IndVAE, MCRL, BetaVAE, iVAE, and SlowVAE, as well as the ablation baselines W/O $e$ and W/O $s$ .

Figure 4: $R^{2}$ scores for comparison methods.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Table of notions
Variables
$\displaystyle{\mathbf{x}}\in\mathbb{R}^{K}$	Observations	$\displaystyle\hat{{\mathbf{x}}}\in\mathbb{R}^{K}$	Reconstructions
$\displaystyle\kappa$	$\|\|L_{{\mathbf{x}}_{\mathbf{a}}\|{\mathbf{z}}}\|\|_{op}\|\|L^{-1}_{{\mathbf{x}}_{\mathbf{a}}\|{\mathbf{z}}}\|\|_{op}$	$\displaystyle\varepsilon$	Identifiability error
$\displaystyle{\mathbf{z}}\in\mathbb{R}^{N}$	Latent variables	$\displaystyle\hat{{\mathbf{z}}}\in\mathbb{R}^{N}$	Latent variable estimations
$\displaystyle\epsilon$	True dependent noise term	$\displaystyle\hat{\epsilon}$	Estimation of $\epsilon$
$\displaystyle\eta$	Auxiliary variable	$\displaystyle\hat{\eta}$	Estimation of $\eta$

Appendix

Table 5

Configuration	Description	Output dimensions
Encoder $q\psi$
Input: ${\mathbf{x}}$		$\text{BS}\times K$
Dense	128 neurons, LeakyReLU	$\text{BS}\times 128$
Dense	128 neurons, LeakyReLU	$\text{BS}\times 128$
Dense	Output embeddings	$\text{BS}\times 2N$
Bottleneck	Compute mean and variance of posterior	$\mu_{\mathbf{z}}$ , $\sigma_{\mathbf{z}}$

Appendix

Table 1: The details of our network architectures for the experiment on MSTM-17 dataset, where BS means batch size,

N=32

M=32

and

K=1280

Figure 5: Visualization of the data generating process for Person Classification task

Methods	Acc
AGW ( Ye et al., 2021 )	85.5 $\pm$ 1.2
TransReID ( He et al., 2021 )	87.8 $\pm$ 0.5
CLIPReID ( Li et al., 2023 )	90.1 $\pm$ 0.3
GTL Yang et al. (2025)	91.5 $\pm$ 1.2
MCRL ( Sun et al., 2025 )	92.6 $\pm$ 0.9
IndVAE ( Hu and Schennach, 2008 )	93.1 $\pm$ 0.5

Appendix

Table 2: Comparison of Top-1 Accuracy on MSMT17 dataset

$N$	IndVAE	Ours
$8$	$0.64\pm 0.06$	$0.80\pm 0.02$
$12$	$0.51\pm 0.03$	$0.68\pm 0.05$
$18$	$0.47\pm 0.04$	$0.61\pm 0.05$

Appendix

Table 3: MCC on the synthetic dataset for increasing latent dimensionality

N

Hyperparameter	Value	MCC
$\lambda$	$1$	$0.87\pm 0.04$
	$0.1$	$0.82\pm 0.03$
	$0.01$	$0.70\pm 0.07$
	$10$	$0.68\pm 0.05$
$\beta_{1}$	$0.02$	$0.87\pm 0.04$
	$1$	$0.73\pm 0.02$

Appendix

Table 4: Sensitivity of MCC to regularization hyperparameters.

Methods	Acc
AGW ( Ye et al., 2021 )	$87.6\pm 0.8$
TransReID ( He et al., 2021 )	$90.2\pm 0.9$
CLIPReID ( Li et al., 2023 )	$91.7\pm 1.1$
GTL Yang et al. (2025)	$93.9\pm 0.5$
MCRL ( Sun et al., 2025 )	$94.5\pm 1.0$
IndVAE ( Hu and Schennach, 2008 )	$95.8\pm 0.8$

Appendix

Table 5: Comparison of Top-1 Accuracy on the RobotPKU dataset.

Methods	Acc
LDP-net Zhou et al. (2023)	$91.7\pm 1.1$
Style Fu et al. (2023)	$92.8\pm 0.8$
CLIPReID ( Li et al., 2023 )	$94.1\pm 1.0$
GTL Yang et al. (2025)	$95.7\pm 0.4$
MCRL ( Sun et al., 2025 )	$96.4\pm 0.8$
IndVAE ( Hu and Schennach, 2008 )	$96.8\pm 0.5$

Appendix

Table 6: Comparison of Top-1 Accuracy on the SYSU-MM01 dataset.

Figure 6: Visualization of the data generations of Eq. 52 .

Method	$A\rightarrow C$	$A\rightarrow P$	$A\rightarrow R$	$C\rightarrow A$	$C\rightarrow P$	$C\rightarrow R$	$P\rightarrow A$	$P\rightarrow C$	$P\rightarrow R$	$R\rightarrow A$	$R\rightarrow C$	$R\rightarrow P$
CLIP ( Radford et al., 2021a )	51.7	81.5	82.3	71.7	81.5	82.3	71.7	51.7	82.3	71.7	51.7	81.5
DAPrompt ( Ge et al., 2023 )	54.1	84.3	84.8	74.4	83.7	85.0	74.5	54.6	84.8	75.2	54.7	83.8
ADCLIP ( Singha et al., 2023 )	55.4	85.2	85.6	76.1	85.8	86.2	76.7	56.1	85.4	76.8	56.1	85.5
UniMoS ( Li et al., 2024a )	59.5	89.4	86.9	75.2	89.6	86.8	75.4	58.4	87.2	76.9	59.5	89.7
Ours	64.1	92.7	88.5	79.2	91.0	89.5	77.9	60.2	89.6	78.4	59.9	91.0

Appendix

Table 7: Results on Office-Home.

Method	Plane	Bicycle	Bus	Car	Horse	Knife	Motorcycle	Person	Plant	Skateboard	Train	Truck
CLIP ( Radford et al., 2021a )	98.2	83.9	90.5	73.5	97.2	84.0	95.3	65.7	79.4	89.9	91.8	63.3
DAPrompt ( Ge et al., 2023 )	97.8	83.1	88.8	77.9	97.4	91.5	94.2	79.7	88.6	89.3	92.5	62.0
ADCLIP ( Singha et al., 2023 )	98.1	83.6	91.2	76.6	98.1	93.4	96.0	81.4	86.4	91.5	92.1	64.2
UniMoS ( Li et al., 2024a )	97.7	88.2	90.1	74.6	96.8	95.8	92.4	84.1	90.8	89.0	91.8	65.3
Ours	98.5	90.1	93.7	76.8	98.7	97.4	96.0	87.2	92.0	93.4	94.2	67.5

Appendix

Table 8: Results on VisDA-2017.

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Diverse Dictionary Learning

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Unsupervised Dictionary Learning Identifiability Latent Variable Learnability Inductive Bias

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Given only observational data

X = g(Z)

, where both the latent variables

Z

and the generating process

g

are unknown, recovering

Z

is ill-posed without additional assumptions. Existing methods often assume linearity or rely on auxiliary supervision and functional constraints. However, such assumptions are rarely verifiable in practice, and most theoretical guarantees break down under even mild violations, leaving uncertainty about how to reliably understand the hidden world. To make identifiability actionable in the real-world scenarios, we take a complementary view: in the general settings where full identifiability is unattainable, what can still be recovered with guarantees, and what biases could be universally adopted? We introduce the problem of diverse dictionary learning to formalize this view. Specifically, we show that intersections, complements, and symmetric differences of latent variables linked to arbitrary observations, along with the latent-to-observed dependency structure, are still identifiable up to appropriate indeterminacies even without strong assumptions. These set-theoretic results can be composed using set algebra to construct structured and essential views of the hidden world, such as genus-differentia definitions. When sufficient structural diversity is present, they further imply full identifiability of all latent variables. Notably, all identifiability benefits follow from a simple inductive bias during estimation that can be readily integrated into most models. We validate the theory and demonstrate the benefits of the bias on both synthetic and real-world data.

Abstract

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